Arrow’s Paradox: The Enigma of Democratic Decision-Making
Don’t you think it is a struggle to reach a collective decision? How often have you participated in a group decision that left you feeling a little let down?
Kenneth J. Arrow, an American economist, noticed the challenges of translating individual preferences into a collective decision. He then stated that it is impossible to create a voting system that follows all the fundamental principles of fair voting. When I say ‘voting’, I mean any collective decision that you reach via a vote, whether it is deciding what to eat in a restaurant to who will govern the state.
So, you may ask, what are the fundamental principles of fair voting?
1. The first one is no dictatorship. This means that the voting system should not be controlled by a single person or a small group.
2. The second is unanimity or Pareto efficiency. This means that in the voting system if all the individuals prefer A over B, the collective decision should also reflect that. For example, if you and your friends are out for dinner and everybody likes pizza more than burgers, then the voting system chosen to determine the outcome should be able to reflect that pizza is the unanimous choice. Quite obvious, right?
3. The third is non-imposition. In simple terms, it means that any possible outcome could in fact become the final outcome if people vote that way. This principle makes sure that the voting system isn’t biased towards certain outcomes. Consider a class trip. Imagine the students have to vote on one of three destinations. In this scenario, each location has a potential to be the ‘winner’ depending on the student vote.
4. The final one is independence of irrelevant alternatives. This means that a preferred order should not change if a choice is removed or added. For example, let’s assume that two people are in a race to become the next class president. The class is divided 60–40 between the two. Now if we add another candidate (who has some ridiculous ideas), who has no chance of winning, this shouldn’t change the outcome between the remaining two candidates.
Arrow proved that no voting system can satisfy all the above criteria. This is called the Arrow’s Paradox or Arrow Impossibility theorem or The General Possibility theorem.
Let us now observe Arrow’s Paradox with some examples.
1. Let us assume that there are three choices, A, B and C. The total number of people voting for these choices is also three. Let’s call them Tom, Dick and Harry. They have been told to rank A, B and C in order of their preference. After voting, the following outcome was observed.
From here, first, let’s check the rankings and see how many people prefer A over B. Clearly, it is in the case of Tom (1>2) and Harry (2>3). Thus, the answer is 2. Two out of three people means that a majority have this preference.
Then let us check how many people prefer option B over C. Again, the answer is 2 since Tom (2>3) and Dick (1>2) have this preference. This is again a majority preference.
Thus, we can conclude that the majority of people prefer A to C since the majority of people prefer A to B and B to C.
A > B and B > C, thus A > C
Now here comes the fun part. If we look at the table again and try to analyse how many people prefer C to A, we see that the answer is again 2 as Dick (2>3) and Harry (1>2) have this preference. This is again a majority.
This is the paradox, where we get a system where the majority prefers A > C and also the majority prefers C > A which is quite impossible.
2. Let us take the example of class presidency once again. For this example, we will assume that Tom, Dick and Harry are the candidates and there are 100 voters. They have pledged to vote fairly and have vowed to vote only for the candidate that they deem worthy of the title. There was tension in the air when the voting began, where, they ranked Tom, Dick and Harry in order of their preferences. After completing the necessary procedures, the results were displayed.
Here is a summary of the results:
So, Tom wins. Easily. The majority of the class thinks that Tom should be the president. Now let us assume that Dick decides to drop off the race. Since Dick is no longer an option, the poll should look like this:
Why so?
In both the above scenarios, 39 people and 14 people have chosen Harry over Tom and when Dick is no longer a participant, the majority has been reversed! This violates property 4 i.e., independence of irrelevant alternatives and hence proves Arrow’s Impossibility theorem.
3. Let’s now look at a last and final example. Again, let us assume that Tom, Dick and Harry have decided to fight for the coveted position of the class president. After the fiasco with the last voting method, they have decided to adopt score voting. It is a system where the voters assign scores to each candidate on a scale of 0–10. After all votes are cast, the scores assigned to each candidate are tallied. The candidate with the highest total score is declared the winner of the election.
If a group of voters decide to deliberately skew the scores in favour of or not in favour of a certain candidate, it can lead to the winning or losing of a candidate.
In the class, 10 voters who preferred Harry over the other two, decided to give Harry a score of 10 while giving a minimum score to the other two. This might lead to Harry winning the election and proving Arrow’s Paradox while violating the no dictatorship property of fair voting.
In real life, Arrow’s Paradox has significant implications. In election systems, it highlights that there will always be some form of trade-off between fairness, efficiency, and representation. In situations where collective decisions need to be made depending on individual preferences like policy-making or resource allocation, a similar situation would arise. It is still a hot topic in social choice theory and people are still engaging in discussions about the nature of collective decision-making.
Overall, Arrow’s paradox serves as a reminder that no voting system is perfect and there will always be some kind of flaw. Any decision-making is complex and often involves compromise and negotiation. Let us hope for a future where such a voting system is developed which can finally disprove Arrow’s paradox and we could then enjoy a fair and impartial procedure. Until then all we have is the collective integrity to rely upon.
